A London council's plan to move housing benefit claimants to Stoke-on-Trent will increase pressure on already vulnerable neighbourhoods and could prompt an increase in "divisive rightwing extremism", according to one housing association asked by the council to accommodate 500 families.

Newham council argues that it can no longer afford to house tenants on its waiting list in private accommodation. It says the rise in rents caused by the Olympics and the demand for housing from young professionals has seen rents rocket in east London.

The gap between market rents and the housing allowance is too big, it says, following a central government cap on housing benefit payments. From January, councils have been writing to claimants telling them of shortfalls in rent that they will have to make up.

Newham, one of the most economically deprived local authorities in the country, which legally must house claimants, says it has been forced to look "further afield for alternative supply" of affordable housing.

Gill Brown, chief executive of Brighter Futures, wrote to local MPs warning of the consequences if "London boroughs are allowed to export their most vulnerable and challenging families to cities like Stoke-on-Trent".

She likened it to the experiences of a decade ago when a Home Office dispersal programme moved thousands of refugees into privately owned properties in north Staffordshire.

"The result was huge unplanned pressure on local services, the collapse of already vulnerable neighbourhoods and the rise of divisive rightwing extremism," she said.

A Labour MP warned that Newham's move was the "tip of the iceberg".

Karen Buck, the Labour MP who was passed the letter, said it showed that ministers were wrong when they claimed nobody would need to move as landlords would drop rents to accommodate welfare cuts. Instead, she said, "there was increasing evidence that local authorities are not able to meet their responsibilities. We see homelessness rising, rents rising and this is a big problem for a government which claimed that none of this would happen."

Sir Robin Wales, the mayor of Newham, rejected the idea that the borough was merely trying to shift problem families to another area.

"No, we've looked at it. And we've done the best we can," he told BBC Radio 4's Today programme.

Brighter Futures was among 1,100 organisations Newham had written to in an effort to find housing for people on an ever-growing list of those needing accommodation, he said.

"We know it is very hard to get property in this borough, and in the East End of London," he said. "We're not looking to push people all to one place, we're looking for the best possible solution for our citizens."

Wales said it wasn't a policy he particularly wanted.

The housing minister, Grant Shapps, accused Newham of overstating the problem and "playing politics". He told Today: "I'm not saying the BBC has been hoodwinked on this, but you have to factor in that it's local election time, and this is a Labour council."

Shapps defended the idea of the housing benefit cap, which gives maximum allowable rents for properties, for example £400 a week for a four-bedroom property. "It can't be right to have people on housing benefit living on streets which hard-working families cannot afford to live on."

Shapps said a search of property websites showed hundreds of homes to rent in Newham. But Wales said many of the landlords of these properties would not accept people on housing benefit.

• This article was amended on 24 April 2012. It incorrectly stated that the housing benefit cap allowed a maximum payment of £26,000 a year, irrespective of family size or local rents. This has been corrected.